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1 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Simultaneous Confidence Regions corresponding to Holm’s Step-down MTP (and other CTPs) Olivier Guilbaud AstraZeneca R&D, Sweden

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2 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Outline Bonferroni Confidence Regions Elements of Proposed Simultaneous Confidence Regions The Simultaneous Confidence Regions Illustration Extensions Nice Reduction Property

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3 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Bonferroni Confidence Regions Estimated quantities (m specified): Marginal -Confidence Regions (m specified): Simultaneous ( )-Confidence Regions: Bonferroni m -adjustment Nice properties: Flexible, Generally valid (if marginal regions are valid) No restriction to particular kinds/dimensions of i s and C i, s

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4 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Can other simultaneous confidence regions based on marginal confidence regions be constructed that share these nice properties ? YES, and Bonferroni regions constitute a special case in a class of such regions.

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5 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements of Proposed Simultaneous Confidence Regions Estimated quantities (m specified): Marginal -Confidence Regions (m specified): Target Regions of interest (m specified): Aim is to ”show”, if possible, that i R i (target assertion) Additional

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6 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements … (cont.) Examples with i and Target region R i for i : Show, i.e., ”superiority” or ”non-inferiority” Show, i.e., ”inequality” Show, i.e., ”equivalence” using appropriate marginal confidence regions C i, s for i s No restriction to particular kinds/dimensions of i s, R i s, or C i, s

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7 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements … (cont.) Raw p-value p i = p i (data) : p i = ( infimum of levels ' for which the test rejects H i ) Marginal Confidence-Region ' Test of H i : i R i to Show H i c : i R i : Connection with Holm’s (1979) Step-down MTP through

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8 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 (Brief Refresher) Ordered p-values: p (1) p (2) … p (m) ; Corresp Hs: H (1), H (2), …, H (m) Bonferroni-Holm’s (1979) MTP with multiple-level : Reject successively H (1), H (2), …, H (m), as long as p (1) /m, p (2) /(m-1), …, p (m) /1 ; Stop at first > ! Sture Holm

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9 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Elements … (cont.) Can be arbitrarily chosen. Can be chosen to sharpen inferences about 1, …, m Given , introduce I Reject ( set of 1 i m of H i s rejected by Holm at multiple level ) I Accept 1, 2, …, m I Reject (i) : the 2 index sets (ii) : additional Estimated quantity & Marginal Confidence Region

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10 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 The Simultaneous Confidence Regions For 1 i m+1, define : Main Result : Bonferroni |I Accept | -adjustment of marginal conf region for i Reflects by how much/little one missed the Target assertion ” i R i ” Useful to sharpen inferences for 1 i m

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11 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Nice Reduction Property What happens if all target regions R 1, …, R m are chosen to be empty ? Reduction to m ordinary Bonferroni Conf Regions for 1, …, m ! because : Only this is informative Empty

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12 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Illustration Estimated quantities in , , e.g. Differences of Trt-means Show, if possible, that each ! That is, each. Marginal 1-sided -Confidence Regions ( t or W based t or W p-values for H 0 : i 0 vs. H 0 c : i > 0 ) Possible realization of Conf Regions with m 5 : 0 Holm non-rejections (Bonf 2 -adjustm) Holm rejections (Target assertions) ”extra free information”

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13 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Illustration (cont.) (Reasonable if scales of are equal or sufficiently similar) Possible choice of additional and to sharpen inferences : Rectangular region Sharpening of Conf Regions with m 5 if occurs : 0 0 instead of L min

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14 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Extension 1: Holm with Weights Holm’s (1979) MTP based on p 1, …, p m and given weights 1, …, m > 0 For 1 i m+1 : Holm rejection index set

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15 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Extension 2: Hommel, Bretz & Maurer (2007) class of MTPs CTP with Bonferroni test of H I using certain w i (I)s, i I, I {1, …, m}. (Fixed-Seq MTP, Holm’s MTP, Gatekeeping MTPs, Fallback MTP, …) For 1 i m+1, again : Weights w i (I) are such that for each non-empty I {1, …, m} : H-B-M rejection index set

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16 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Final Comments Flexible (no restriction concerning kinds/dim of i s, R i s and C i, s) Multi-dimensional i s and R i s can be combined with ”marginal” simultaneous CIs within i s (e.g. Hsu-Berger CIs for i -components): Generally valid (if marginal confidence regions are valid) Intuitively appealing Simple to implement Leads e.g. to Simultaneous Confidence Regions corresponding to the Bonferroni-Holm MTP for m families of hypotheses (Bauer et al appendix, Bauer et al. 2001) with Extra ”free” Information m=2 : Superiority vs. PlaceboNon-inferiority vs. Placebo

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17 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Selected References Aitchinson, J. (1964). Confidence-region Tests. Journal of the Royal Statistical Society, Ser. B, 26, Hsu, J. C. and Berger, R. L. (1999). Stepwise Confidence Intervals Without Multiplicity Adjustment for Dose-response and Toxicity Studies. Journal of the American Statistical Association 94, Bauer, P., Brannath, W., and Posch, M. (2001). Multiple Testing for Identifying Effective and Safe Treatments. Biometrical Journal 43, Bauer, P., Röhmel, J., Maurer, W., and Hothorn, L. (1998). Testing Strategies in Multi-dose Experiments Including Active Control. Statistics in Medicine 17, Holm, S. (1979). A Simple Sequentially Rejective Multiple Test Procedure. Scandinavian Journal of Statistics 6, Hommel, G., Bretz, F., and Maurer, W. (2007). Powerful Short-Cuts for Multiple Testing Procedures with Special Reference to Gatekeeping Strategies. Statistics in Medicine (in press).

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18 O Guilbaud, MCP2007, July 8-11, Vienna, v.1 Summary Bonferroni Confidence Regions Elements of Proposed Simultaneous Confidence Regions The Simultaneous Confidence Regions Illustration Extensions Nice Reduction Property

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